Local convergence of inexact methods under the Hِlder condition

We study the convergence properties for some inexact Newton-like methods including the inexact Newton methods for solving nonlinear operator equations on Banach spaces. A new type of residual control is presented. Under the assumption that the derivative of the operator satisfies the Hölder condition, the radius of convergence ball of the inexact Newton-like methods with the new type of residual control is estimated, and a linear and/or superlinear convergence property is proved, which extends the corresponding result of [B. Morini, Convergence behaviour of inexact Newton methods, Math. Comput. 68 (1999) 1605–1613]. As an application, we show that the inexact Newton-like method presented in [R.H. Chan, H.L. Chung, S.F. Xu, The inexact Newton-like method for inverse eigenvalue problem, BIT Numer. Math. 43 (2003) 7–20] for solving inverse eigenvalue problems can be regarded equivalently as one of the inexact Newton-like methods considered in this paper. A numerical example is provided to illustrate the convergence performance of the algorithm.